Question

Evaluate the integral $\int_0^2 e^{x^3}x^2dx$.

Answer 1

$$\int e^{x^3}x^2dx$$

$u=x^3, du=3x^2dx$

$$\int e^{x^3}x^2dx=\dfrac{1}{3}\int e^{u}du=\dfrac{1}{3}e^u+C$$

$$=\dfrac{1}{3}e^{x^3}+C$$

$$\displaystyle\int_{0}^{2}e^{x^3}x^2dx=[\dfrac{1}{3}e^{x^3}]\begin{matrix} 2 \\ 0 \end{matrix}=\dfrac{1}{3}e^{8}-\dfrac{1}{3}$$